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Iterative Aggregation Disaggregation

Explore the process of webpage ranking using Google's page ranking algorithm. Learn about the iterative aggregation disaggregation method and its application in conditioning the matrix for effective page ranking.

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Iterative Aggregation Disaggregation

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  1. Iterative Aggregation Disaggregation Nicole Typaldos Missouri State University

  2. Graph Web Graph Matrix Pagerankvector Process of Webpage ranking

  3. Google’s page ranking algorithm

  4. Conditioning the matrix H • Definitions: • Reducible: if there exist a permutation matrix Pnxn and an integer 1 ≤ r ≤ n-1 such that: • otherwise if a matrix is not reducible then it is irreducible • Primitive: if and only if Ak >0 for some k=1,2,3…

  5. Example Set Up

  6. Example Continued

  7. Example Continued

  8. The Google Matrix • Where • ℮ is a vector of ones • U is an arbitrary probabilistic vector • a is the vector for correcting dangling nodes > 0

  9. Example Continued

  10. Different Approaches • Power Method • Linear Systems • Iterative Aggregation Disaggregation (IAD)

  11. Linear Systems andDangling Nodes • Simplify computation by arranging dangling nodes of H in the lower rows • Rewrite by reordering dangling nodes • Where is a square matrix that represents links between nondangling nodes to nondangling nodes; is a square matrix representing links to dangling nodes

  12. Rearranging H

  13. Exact aggregation disaggregation • Theorem • If G transition matrix for an irreducible Markov chain with stochastic complement: is the stationary dist of S, and is the stationary distribution of A then the stationary of G is given by:

  14. Approximate aggregation disaggregation • Problem: Computing S and is too difficult and too expensive. So, • Ã= • Where A and à differ only by one row • Rewrite as: • Ã=

  15. Approximate aggregation disaggregation • Algorithm • Select an arbitrary probabilistic vector and a tolerance є • For k = 1,2, … • Find the stationary distribution of • Set • Let • If then stop Otherwise

  16. Combined methods • How to compute • Iterative Aggregation Disaggregation combined with: • Power Method • Linear Systems

  17. With Power Method • = Ã • Ã is a full matrix • = =

  18. With Power Method • Try to exploit the sparsity of H • solving = Ã • Exploiting dangling nodes:

  19. With Power Method • Try to exploit the sparsity of H • Solving = Ã • Exploiting dangling nodes:

  20. With Linear Systems • = Ã • After multiplication write as: • Since is unknown, make it arbitrary then adjust

  21. With Linear Systems • Algorithm (dangling nodes) • Give an initial guess and a tolerance є • Repeat until • Solve • Adjust

  22. References • Berry, Michael W. and Murray Browne. Understanding Search Engines: Mathematical Modeling and Text Retrieval. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2005. • Langville, Amy N. and Carl D. Meyer. Google's PageRank and Beyond: The Science of Search Engine Rankings. Princeton, New Jersey: Princeton University Press, 2006. • "Updating Markov Chains with an eye on Google's PageRank." Society for Industrial and Applied mathematics (2006): 968-987. • Rebaza, Jorge. "Ranking Web Pages." Mth 580 Notes (2008): 97-153.

  23. Iterative Aggregation Disaggregation Nicole Typaldos Missouri State University

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